论文标题
部分可观测时空混沌系统的无模型预测
Simple Continued Fractions an Approach for High School Students
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
This paper aims to introduce high school students to the intriguing world of continued fractions, a mathematical concept that provides a unique representation of numbers. The study focuses on the exploration and development of the fundamental properties of both Finite and Infinite Continued Fractions. It further delves into the computation of quadratic numbers using given periodic continued fractions and the concept of conjugate quadratic numbers. A significant part of the paper is dedicated to the approximation of real numbers and the convergence properties of continued fractions. The study of continued fractions offers a profound understanding of the intricate relationships within number systems, a key emphasis in contemporary mathematics education. The paper is designed to be engaging and interactive, fostering a fun and stimulating learning environment. By the end of this study, students will have gained a comprehensive understanding of continued fractions, their properties, and their applications, thus enhancing their mathematical proficiency and problem-solving skills. This paper serves as a stepping stone for students to explore more complex mathematical concepts and theories, fostering a deeper appreciation for the subject.
